kissing-11

Metric points, maximize · Best known 593 (AlphaEvolve 2025) · Code · All example problems

hillclimb problem get kissing-11      # copies the problem into hillclimb/problems/
hillclimb verify kissing-11           # scores the floor; spread 0, the verifier is exact
hillclimb run kissing-11 --budget 1h

The problem

Find as many nonzero integer points as possible in Z^11 such that every pairwise distance is at least the largest norm among the points:

min over i != j of ||p_i - p_j|| >= max over i of ||p_i||

Unit spheres centred at 2 p_i / ||p_i|| then all touch the unit sphere at the origin without overlapping, so the number of points N is a lower bound on the kissing number in dimension 11. The score is N.

Constraints (verified programmatically, exactly, on squared integer norms):

• every coordinate is an integer with |c| <= 100000000
• no point is the origin, no two points are equal
• min_(i != j) ||p_i - p_j||^2 >= max_i ||p_i||^2
• at most 2000 rows

The best known configuration in dimension 11 has 593 points (AlphaEvolve 2025, arXiv:2506.13131 Appendix B.11; the previous record was 592, Ganzhinov 2022). The score is an integer count, so most edits are plateaus: a candidate that keeps N is a tie, not a loss, and progress comes in unit steps. Good approaches: start from lattice shells (the 220 points ±e_i ± e_j of D_11 already satisfy the condition), scale a configuration up so there is integer room to insert extra points, and then search for insertions/replacements that keep the min-distance inequality, repairing the worst pair after each move. The check is cheap (a Gram matrix), so many local moves fit in the time budget. numpy and scipy are available.

Submission format

Write submission.csv in the working directory with the header id,c0,c1,c2,c3,c4,c5,c6,c7,c8,c9,c10 and one row per point (id = 0..N-1, any N >= 1, integer coordinates), like sample_submission.csv (a weak valid baseline: the 22 points ±e_i).

Scoring

The orchestrator runs problem/verify.py after your script finishes. The verifier validates the configuration and prints val_score: <N> (0 if invalid). Higher is better.

There is no train/test data; this is a pure construction problem. Keep total runtime well within the execution time limit.